{"id":"https://openalex.org/W7150758407","doi":"https://doi.org/10.48550/arxiv.2604.02797","title":"A Non-compact Positivity-Preserving Numerical Scheme for Elliptic Differential Equations Based on Mathematical Expectation","display_name":"A Non-compact Positivity-Preserving Numerical Scheme for Elliptic Differential Equations Based on Mathematical Expectation","publication_year":2026,"publication_date":"2026-04-03","ids":{"openalex":"https://openalex.org/W7150758407","doi":"https://doi.org/10.48550/arxiv.2604.02797"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.02797","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.02797","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.02797","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5123302473","display_name":"Haoran Xu","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xu, Haoran","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133026819","display_name":"Kunyang Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Kunyang","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5133047112","display_name":"Xingye Yue","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yue, Xingye","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5123302473"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.29339998960494995,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.29339998960494995,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.13600000739097595,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.09399999678134918,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.4966999888420105},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.43220001459121704},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4246000051498413},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.3939000070095062},{"id":"https://openalex.org/keywords/neumann-boundary-condition","display_name":"Neumann boundary condition","score":0.37929999828338623},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.3765000104904175},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.3718000054359436},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.3513000011444092},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.3456999957561493}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7684000134468079},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5274999737739563},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.4966999888420105},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.43220001459121704},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4246000051498413},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.3939000070095062},{"id":"https://openalex.org/C163681178","wikidata":"https://www.wikidata.org/wiki/Q1149279","display_name":"Neumann boundary condition","level":3,"score":0.37929999828338623},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.3765000104904175},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.3718000054359436},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3560999929904938},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.3513000011444092},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.3456999957561493},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3163999915122986},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.3158000111579895},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.3095000088214874},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3046000003814697},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.29820001125335693},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.29440000653266907},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.2939999997615814},{"id":"https://openalex.org/C189973286","wikidata":"https://www.wikidata.org/wiki/Q176695","display_name":"Markov property","level":4,"score":0.2915000021457672},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.28690001368522644},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.2858999967575073},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C106666656","wikidata":"https://www.wikidata.org/wiki/Q6771323","display_name":"Markov kernel","level":5,"score":0.265500009059906},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.2554999887943268}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.02797","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.02797","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.02797","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.02797","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0],"propose":[1],"a":[2,23,28,39,99,110],"novel":[3],"non-compact,":[4],"positivity-preserving":[5,138],"scheme":[6,41],"for":[7,61],"linear":[8],"non-divergence":[9],"form":[10],"elliptic":[11],"equations.":[12],"Based":[13],"on":[14],"the":[15,18,44,74,84,147],"Feynman--Kac":[16],"formula,":[17],"solution":[19],"is":[20,59,77,87,115],"represented":[21],"as":[22],"conditional":[24],"expectation":[25,45],"associated":[26],"with":[27,46,65],"diffusion":[29,63],"process.Instead":[30],"of":[31,83,91],"using":[32],"compact":[33],"Markov":[34],"chain":[35],"approximations,":[36],"we":[37,97],"construct":[38],"wide-stencil":[40],"by":[42],"approximating":[43],"carefully":[47],"designed":[48],"transition":[49],"probabilities,":[50],"ensuring":[51],"both":[52],"consistency":[53],"and":[54,137],"positivity":[55],"preservation.":[56],"The":[57,131],"method":[58],"effective":[60],"anisotropic":[62],"problems":[64],"mixed":[66],"derivatives,":[67],"where":[68],"classical":[69],"schemes":[70,133],"typically":[71],"fail":[72],"unless":[73],"covariance":[75],"matrix":[76],"diagonally":[78],"dominant.":[79],"A":[80],"key":[81],"feature":[82],"proposed":[85],"framework":[86],"its":[88],"robust":[89],"treatment":[90],"boundary":[92,153],"conditions.":[93],"For":[94,107],"Dirichlet":[95],"boundaries,":[96,109],"introduce":[98],"quadtree-based":[100],"non-uniform":[101],"stopping-time":[102],"strategy,":[103],"achieving":[104,128],"$O(h)$":[105,129],"accuracy.":[106,130],"Neumann":[108],"discrete":[111],"specular":[112],"reflection":[113],"mechanism":[114],"employed,":[116],"yielding":[117],"$O(h^{1/2})$":[118],"convergence.":[119],"Periodic":[120],"boundaries":[121],"are":[122,134],"handled":[123],"through":[124],"modular":[125],"wrapping,":[126],"also":[127],"resulting":[132],"unconditionally":[135],"stable":[136],"due":[139],"to":[140],"their":[141],"probabilistic":[142],"structure.":[143],"Numerical":[144],"experiments":[145],"confirm":[146],"theoretical":[148],"convergence":[149],"rates":[150],"under":[151],"all":[152],"conditions":[154],"considered.":[155]},"counts_by_year":[],"updated_date":"2026-04-07T06:06:30.997549","created_date":"2026-04-07T00:00:00"}